A Fast Eigenvalue Approach for Solving the Trust Region Subproblem with an Additional Linear Inequality

TL;DR

This paper introduces an efficient eigenvalue-based method for solving the extended trust region subproblem with an additional linear inequality, leveraging duality and eigenvalue reformulation for large-scale problems.

Contribution

It reformulates the extended trust region subproblem as a two-parameter eigenvalue problem and provides a necessary and sufficient condition for strong duality, enabling efficient solutions.

Findings

The proposed method effectively solves large-scale eTRS problems.

Numerical experiments demonstrate the efficiency and accuracy of the approach.

The approach identifies conditions under which strong duality holds.

Abstract

In this paper, we study the extended trust region subproblem (eTRS) in which the trust region intersects the unit ball with a single linear inequality constraint. By reformulating the Lagrangian dual of eTRS as a two-parameter linear eigenvalue problem, we state a necessary and sufficient condition for its strong duality in terms of an optimal solution of a linearly constrained bivariate concave maximization problem. This results in an efficient algorithm for solving eTRS of large size whenever the strong duality is detected. Finally, some numerical experiments are given to show the effectiveness of the proposed method.